some of the ideas that gunnarsen told me about related to basic algebraic number theory topics such as ideal class groups of quadratic number fields, and certain "graphical" methods of demonstrating them. i haven't really grasped yet what these graphical methods are doing, but they vaguely reminded me at a naive visual (and/or tactile, etc) level of methods that i developed myself in trying to understand and explain approximately the same topics. so i started to try to remind myself of the methods that i developed, which is what the unfinished proof below is about. (later i started to think that the visual resemblances that i noticed were probably just superficial; in any case i still want to understand gunnarsen's methods and how they relate to various ideas that i've explored.)

here's a "proof" that a prime p of the form 4n+1 is a sum of two squares:

(i'll illustrate the proof with pictures corresponding to the case p=5.)

take a square grid:

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(picture may have distorted aspect ratio)

and inside of it mark off a similar but sparser grid with nodes spaced p units apart:

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now in "clock arithmetic" modulo p, there must exist a square root of -1, because the multiplicative group is cyclic of order 4n ...